Figure 4.5: Phonon density of states for austenitic ...

CHAPTER 4. LATTICE DYNAMICS OF AUSTENITIC STAINLESS STEELS 41

Figure 4.5: Phonon density of states for austenitic stainless steel Fe-18Ni-12Cr-2Mo, (•)experimental data, (—) multiphonon contribution and (4) DOS after multiphonon correc-tion, < u2

α >=0.00403 A2.

Figure 4.6: Phonon density of states for austenitic stainless steel Fe-18Ni-10Cr, (•) exper-imental data, (—) multiphonon contribution and (4) DOS after multiphonon correction,< u2

α >=0.00392 A2.

41


Figure 4.7: Phonon density of states for austenitic stainless steel (•) Fe-18Ni-12Cr-2Mo,(4) Fe-18Ni-16Cr-10Mn and Fe-18Ni-10Cr (×) after multiphonon correction.

After the subtraction of the multiphonon contribution (fig. 4.4,4.5,4.6) the data were cor-rected by the resolution function of the spectrometer to enable a comparison with modelcalculations. For the resolution function a gaussian distribution with a FWHM of 2 meVwas used. Data for resolution function including FWHM are obtained from a vanadiumscan.

Figure 4.8: Phonon density of states, (•) experimental data for austenitic stainless steelsFe-18Cr-16Ni-10Mn, (—) comparison with simulated data using the Born-von-Karmanmodel, generalized force with 2 neighbouring shells and (—)with the same model usingcontributions from the resolution function.

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The phonon density of states in Fe-18Cr-16Ni-10Mn, Fe-18Cr-12Ni-2Mn and Fe-18Cr-10Mnshow a great similarity (Fig. 4.7). The derived DOS show a significant difference to theexperimental data, but when the resolution function was introduced the discrepancy issignificantly reduced (fig. 4.8). This behaviour is especially pronounced for the longitudinalmode of vibration (fig. 4.8).

4.4 Thermal properties

From both the INS data on single crystals as well as on polycrystalline samples values wereobtained for the elastic constants and moduli, respectively. On the other hand thermalproperties are much more sensitive to the peak in the DOS than to the low energy limits.Using the relation for the Debye temperature from [Beskrovni 1999] a value of Debyetemperature ΘD=416 K was obtained. The values for the Debye temperature were in goodagreement with experimental data for similar Fe-Cr-Ni alloys [Beskrovni 1999](eq. 2.859and with pure metals Fe, Cr, Ni, Mo. From the DOS the specific heat capacity cV (eq.2.80), the internal energy U(eq. 2.78), the mean squared displacement < ∆x2 >(eq. 3.19)and Debye temperature ΘD(T )(eq. 2.81) can be obtained (table 4.2)

Table 4.2: Debye temperature of some transition metals and some of their alloys (austeniticstainless steels).

system Debye temperature ΘD [K]Fe-Cr-Ni alloys 386-396 [Beskrovni 1999]

Fe 420 [Ashcroft 1976]Ni 375 [Ashcroft 1976]Cr 460 [Ashcroft 1976]Mn 400 [Ashcroft 1976]Mo 380 [Ashcroft 1976]

Fe-18Cr-12Ni-2Mo 406 [this study]Fe-18Cr-16Ni-10Mn 410 [this study]

generalised force model 2nd 418 [this study]generalised force model 2nd 414 [this study]

with electron gas contribution

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Figure 4.9: Specific heat capacity cV (—)calculated from DOS for Fe-18Cr-12Ni-2Moand (—) from the Debye model for Debyetemperature ΘD =406 K.

Figure 4.10: Debye temperature as a func-tion of temperature for the austenitic stainlesssteels (—) Fe-18Cr-16Ni-10Mn and (· · · ) Fe-18Cr-12Ni-2Mo.

Figure 4.11: Internal energy U (—) cal-culated from experimental data for Fe-18Cr-12Ni-2Mo and (—) from the Debyemodel for Debye temperature ΘD =406K.

Figure 4.12: Mean squared displacement < ∆x2 >calculated from experimental data (—) for Fe-18Cr-12Ni-2Mo and (—) from the Debye modelfor Debye temperature ΘD =406 K.

A mean squared < ∆x2 >=0.00412 A2 was obtained, which corresponds to Biso=0.325A2 from the Debye model values for a given Debye temperature ΘD=416 K. Similar datawas also obtained directly from the phonon density of states in the austenitic stainless

44


steels Fe-18Cr-12Ni-2Mo, Fe-18Cr-16Ni-10Mn and Fe-18Cr-10Ni. These values showedgood agreement with other diffraction studies from which values of Biso=0.3-0.4 A2 wereobtained.

4.5 Conclusion

The phonon dispersion curves for the steels Fe-18Cr-12Ni-2Mo and Fe-18Cr-16Ni-10Mn arevery similar (fig. 4.3). The Born-von-Karman model together with the generalized forcematrices (5 neighbouring shells) fits well to the experimental data (Fig. 4.3). Using theBorn-von Karman model with only two neighbouring shells and introducing the electrongas contribution instead, a significant improvement of the fit can be achieved.(Fig. 4.2).With this approach a satisfactory fitting with only few fitting parameters was obtained.From the force constants and applying the Born-von-Karman model the elastic constantsand moduli were obtained (fig. 4.1). The obtained values show good agreement withultrasonic measurements on different Fe-Cr-Ni-Mn systems. For comparison, also ultra-sonic studies in the system Fe-18Cr-12Ni-2Mo [Ledbetter 1981] were used. In the case ofFe-18Cr-16Ni-10Mn [Kim 1994] empirical relations as well as data of ultrasonic studies onpolycrystalline samples were used. To obtain really accurate data for the elastic constantsand moduli, it is necessary to perform ultrasonic studies on single crystal samples.Densities of states for austenitic stainless steels Fe-18Cr-12Ni-2Mo, Fe-18Cr-16Ni-10Mnand Fe-18Cr-10Ni are extremely similar (fig. 4.7). Modelling of DOS based on force con-stants obtained from dispersion curves, show a significant discrepancy (fig. 4.8). Instru-ment resolution function smears the peak profiles, which have been derived from dispersioncurves. When the contribution of the resolution function is also included, it significantlyreduces the difference between the experimental and the modelled data. The differencesbetween different steels are much smaller than those between experiment and modelling.From the DOS the thermal properties were obtained Debye temperature Θ(T ) (fig. 4.10),internal energy (fig. 4.11) and specific heat capacity (fig. 4.9). The agreement for theDebye temperature obtained from a low energy fit of the DOS and the values calculatedfrom the Einstein-Madelung-Sutherland relation (eq. 2.85) is pretty good. These valuesare similar to those reported by [Beskrovni 1999] for other Fe-Cr-Ni-Mn systems basedon inelastic scattering studies (table 4.2). It is observed, that Θ∞ is slightly higher forthe Fe-18Cr-10Ni system, this could be connected to a lower intensity of the longitudinalmodes. The heat capacities of all 3 samples show great similarity to each other and tothe calculations based on the Debye model. Beside this, calculations of the mean squareddisplacement based on the Debye model (fig. 4.12) and directly derived from the den-sity of states are plausible to each other. This result also confirms the correctness of thecalculation procedure.

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CHAPTER 5. LATTICE DYNAMICS OF HYDROGENATED AUSTENITICSTAINLESS STEELS 46

Chapter 5

Lattice dynamics of hydrogenatedaustenitic stainless steels

The results of diffraction are used as the input for performing modelling. Especially thedistribution of hydrogen and the lattice parameters are important. From this data metal-hydrogen distances were obtained.

5.1 Diffraction study

5.1.1 Experiment

The hydrogenations have been carried out in the Institute of Solid State Physics of theRussian Academy of Sciences in Chernogolovka (Moscow district). The samples consistedof stacks of 3 - 6 discs, each of them has 8 mm in diameter and 0.25 mm thickness. Thehydrogen loadings were performed in a high pressure cell at 350 0C for 24 h. Variouspressures - up to 7 GPa - were applied in order to obtain several hydrogen concentrationsup to H/Me≈1. In addition to hydrogenations, also deuterium chargings have been carriedout. After the hydrogen chargings the samples were cooled down to 100 K and stored atliquid nitrogen to prevent the outgassing of hydrogen.In the case of samples with low hydrogen content(H/Me�0.1) loading pressure are 10-20MPa. Samples are hydrogenated and deuterated at the Institute for solid state physics,University of Darmstadt.

Neutron diffraction

The neutron diffraction experiments have been carried out4 at diffractometer E9 at theHahn-Meitner-Institute in Berlin, BT1 at the National Institute of Standards and Tech-nology(NIST) in Gaithersburg as well as D1B in Institute Laue-Langevin (ILL) in Grenoble.Diffraction experiment was performed at low temperature(80 K) to prevent outgassing.

4experiments and Rietveld refinements were performed by Markus Holzel

46


At powder diffractometer D1B Fe-25Cr-20Ni and Fe-18Cr-10Ni were characterised. In thecase of Fe-18Cr-10Ni with low hydrogen content, neutron diffraction was performed attriple axis spectrometer E1(HMI). Neutron with wave length λ=2.431 A, were used withgraphite(002) monochromator and λ=1.231 A, with Ge(311) monochromator. Highly hy-drogenated samples are investigated on the diffractometer E9(HMI), with Ge(711) monochro-mator which allow wavelength of λ=1.3078 A. At the powder diffractometer BT1 Fe-18Cr-10Ni with loading pressures between 3-7 GPa on 80 K were investigated. The monochro-mator is Cu(311) and λ=1.5398 A.

X-ray diffraction

X-ray diffraction was performed5 in samples with very low hydrogen content at room tem-perature with powder diffractometer D500 for the checking of phase composition.The X-ray investigations on samples with high hydrogen content powder diffractometerSTOE STADI P was used. In this experiment as source was used cathode tube withMo-Kα1 and Cu-Kα in the transmission mode. Samples were cooled with gas flow nitrogen.

5.1.2 Result

Diffraction data were analysed with software package [Fullprof 2002].

Figure 5.1: The Rietveld refinement of the powder neutron diffraction data of Fe-25Cr-20Ni( diffractometer E9, λ=1.30747 A).

5experiments and Rietveld refinements were performed by Markus Holzel

47


Figure 5.2: The Rietveld refinement of the powder neutron diffraction data of hydrogenatedFe-25Cr-20Ni with loading pressure of 2.3 GPa ( diffractometer E9, λ=1.30747 A).


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Figure 5.5: The Rietveld refinement of the powder neutron diffraction data of deuteratedFe-25Cr-20Ni with loading pressure of 7.0 GPa ( diffractometer E9, λ=1.30747 A).

49


Figure 5.6: The Rietveld refinement of the powder neutron diffraction data of Fe-25Cr-20Niwith loading pressure of 4.0 GPa ( diffractometer E9, λ=1.30747 A).

Table 5.1: Result of Rietveld refinement for austenitic stainless steel Fe-25Cr-20Ni

hydrogenation weight frac. lattice H(D) occ. in Biso(H)

pressure [GPa] of phases constant (at 80 K) octahed. sites (A2)uncharged 100 % Fm3m a = 3.5826(1) - -

2.3 100 % Fm3m a = 3.6586(1) 35(2) % 1.353.0 100 % Fm3m a = 3.7261(2) 66(3) % 1.447.0 100 % Fm3m a = 3.7798(2) 94(5) % 1.48

Deuteration 100 % Fm3m a = 3.6236(1) 13(1) % 0.964.0 without H 100 % Fm3m a = 3.5824(2) - -

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Figure 5.7: The Rietveld refinement of the powder neutron diffraction data of Fe-18Cr-10Ni( diffractometer E9, λ=1.30747 A).

γ

γ

γ

γ

γ

γ

γ

γ

γγ

γ


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Figure 5.10: The Rietveld refinement of the powder neutron diffraction data of hydro-genated Fe-18Cr-10Ni with loading pressure of 7.0 GPa ( diffractometer E9, λ=1.30747A).

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Figure 5.11: The Rietveld refinement of the powder neutron diffraction data of Fe-18Cr-10Ni pressure of 4.0 GPa ( diffractometer E9, λ=1.30747 A).

Table 5.2: Result of Rietveld refinement for austenitic stainless steel Fe-18Cr-10Ni

hydrogenation weight frac. lattice H occ. in Biso(H)

pressure [GPa] of phases constant (at 80 K) octahed. sites (A2)uncharged 100 % Fm3m a = 3.5800(1) - -

2.3 100 % Fm3m a = 3.6537(1) 30(2) % 1.393.0 84 % Fm3m a = 3.6988(2) 56(2) % 1.41

16 % P63/mmc a = 2.617(2) 71(9)%c = 4.254(3)c/a = 1.626

7.0 Fm3m a = 3.7757(3) 103(9) % 1.48P63/mmc a = 2.6676(8);

c = 4.335(2)c/a = 1.625

4.0 without H 86 % Fm3m a = 3.5785(2) - -11 % P63/mmc a = 2.6676(8);

c = 4.335(2)c/a = 1.625

3 % Im3m a = 2.857(3)

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Simulation are performed to justify the results of Rietveld analysis in hydrogenated austeniticstainless steels. In the fcc structure exist two possible sites for hydrogen atoms: in tetrahe-dral and octahedral voids. Using the relation for Fhkl(eq. 3.7) it is easy to obtain relativeintensity of the diffraction maxima.

Table 5.3: Structure factors |Fhkl|2 for hydrogen in tetrahedral and octahedral interstitials

sites.

H-tetrahedral sites H-octahedral sitesZnS structure random alloy NaCl structure

h,k,l=2n+1 16(b2Me+b2

H) 16b2Me 16(bMe-bH)2

h+k+l=4n 16(bMe+bH)2 16(bMe+bH)2 16(bMe+bH)2

h+k+l=4n+2 16(bMe-bH)2 16(bMe-bH)2

Using a Fullprof program it is possible to perform simulation of neutron diffraction patternfor hydrogen atoms located in octahedral and tetrahedral interstitial sites.

Figure 5.12: Simulated powder neutron diffraction pattern on Fe-25Cr-20Ni with hydrogenlocated in octahedral sites(H/Me=1)

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Figure 5.13: Simulated powder neutron diffraction pattern on Fe-25Cr-20Ni with hydrogenlocated in tetrahedral sites(H/Me=1)

Figure 5.14: Simulated powder neutron diffraction pattern on Fe-25Cr-20Ni with hydrogenlocated in tetrahedral sites(H/Me=1) which build ZnS structure

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5.1.3 Conclusion

From the result of Rietveld refinements for the austenitic stainless steels Fe-18Cr-10Niand Fe-25Cr-20Ni it is straightforward to conclude about phase stability. Increase of thelattice parameter of both steels with increasing amount of hydrogen in interstitial sites.Austenitic stainless steel Fe-25Cr-20Ni(table 5.1) shows great stability on high pressuretreatment up to 4.0 GPA as well as on the hydrogenation in high pressure cell. In the caseof Fe-18Cr-10Ni(table 5.1.2) after hydrogenation under the pressure of 3.0(H/Me∼0.56)and 7.0 GPa(H/Me∼1) hexagonal phase appear. When only high pressure treatment(4.0GPa) is applied appears not only hexagonal but also a minor fraction of body centeredcubic(bcc) α-phase.

5.2 Inelastic neutron scattering

5.2.1 Experiment

Inelastic neutron scattering was performed on polycrystalline samples of hydrogenatedaustenitic stainless steels. In the phonon density of states g(ω) exist two modes: acousticand optic with a gap between them.

Acoustic modes

Acoustic modes have energies similar to non-hydrogenated austenitic stainless steels whichis in the range of 0-40 meV. Due to hydrogenation procedure strength of bonding andupper limit of acoustic modes are reduced. Inelastic neutron scattering was performed6 inFANS(Filter Analyser Neutron Spectrometer) at the National Institute of Standards andTechnology(NIST) in Gaithersburg. This method also has significantly higher statistics.In the energy range of 20-25 meV a strong peak was observed. In this device Cu(220) isused as monochromator. In the case of neutrons which are nonrelativistic particles thefurther relation is valid:

ε =mnv

2

2=

h2

2mn

1

λ2. (5.1)

The monochromated beam always contains λ/2 component. In the case of neutrons withenergy of 25 meV the corresponding λ/2 component has an energy of 100 meV. Neutronwhich lost 25 meV could be scattered from acoustic or optic modes(λ/2 component) ofvibrations. In the experimental data strong overlapping of response of acoustic modes ofthe samples with excitation of optic modes occurs. This overlapping is so strong that it isimpossible to distinguish between them.

6experiments were performed by Markus Holzel

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Figure 5.15: Experimental data for phonon density of states (-·-)austenitic stainless steelFe-25Cr-20Ni without hydrogen, (-·-) Fe-25Cr-20Ni with H/Me=0.35(2), (-·-) Fe-18Cr-10Niwith H/Me=0.56(2) and (-·-) Fe-18Cr-10Ni with H/Me=1.03(9).

Optic modes

Vibration spectra of the hydrogen atoms are located in high energy region, over 100 meV.In this region ~ω � kT (eq. 3.38). Therefore energy loss spectroscopy is used which inthis region has higher intensity. All spectrometers are optimised for high energy regionand so the λ/2 component is reduced. INS was performed7 in FANS at NIST in Gaithers-burg , IN1-BeF in the Institute Laue-Langevin (ILL) in Grenoble and TOF spectrometerTOSCA(Thermal Original Spectrometer with Cylindrical Analysers) in Rutherford Apple-ton Laboratory, Oxfordshire.Filter spectrometers IN1-BeF and FANS measure neutron energy loss spectra with fixedfinal energy. Monochromatisation is performed by Bragg reflection on the monochromatorcrystal. In the case of IN1 spectrometer, neutrons came from the hot source with high flux.Spectrometer IN1-BeF has a beryllium filter which allows passing of neutrons with energyaround 3.5 meV. FANS spectrometer use thermal neutrons and graphite and berylliumfilters. The detection energy is fixed to value of a 1.8 meV.Time of flight spectrometer TOSCA is mounted on the spallation source (ISIS) in theindirect geometry and as analyser graphite is used.

7experiments were performed by Markus Holzel

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5.2.2 Multiphonon correction

To obtain spectra for modelling, the multiphonon part has to be removed. On the spec-tra in the range 80-150 meV in multiphonon spectra exist one main contribution, whichis a convolution of acoustic and optics modes. FANS spectrometer is optimised for datacollection in the high energy range, so acoustics are or not measured at all or have big dis-turbance due to λ/2 contribution. Reliable data for acoustics not exist which are necessaryfor multiphonon correction.The Debye-Waller exponent is proportional to differential cross section. Debye-Waller ex-ponent is mainly determined through acoustic modes. Therefore values obtained fromdiffraction studies are used as starting values for the Debye-Waller exponent. Further re-finement of Debye-Waller exponent was performed during multiphonon correction. Withthe assumption of similarity of acoustic modes of all austenitic stainless steels, density ofstates g(ω) was used as a base of the calculation. The obtained DOS for the austeniticstainless steels was corrected due to lattice expansion by Gruneisen parameter. On thecorrected DOS for acoustic part of spectra its multiphonon contribution was superposed,in this case only first iteration. After that standard procedure for multiphonon correctionwas applied.

Figure 5.16: Optic modes in phonon density of states for hydrogenated austenitic stainlesssteel Fe-18Ni-10Cr(at 7.0 GPa) with H/Me=1.03(9), (-•-) experimental data, (—) multi-phonon contribution and (-4-) DOS after multiphonon correction.

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